Given:
    q_n+1 = [(1 - s)*q _n]/[1 - s*q _n]

Find: An equation for q_n in terms of q₀, s, and n.

Remember the equation for q_n+1  as a function of q_n , s, and n:
    q_n+1 = [(1 - s)*q _n]/[1 - s*q _n]

Well, lets rearrange that into a more suitable form to
    q_n+1 = [(1 - s)*q _n/q _n ] / [(1 - s*q _n)/q _n]                  (Divided numerator and denominator by q _n.)
    q_n+1 = (1 - s) / [1/q _n - s*q _n/q _n]
    q_n+1 = [(1 - s)] / [(1/q _n - s)]
    q_n+1 = [(1 - s)/(1 - s)] / [(1/q _n - s)/(1 - s)]                (Divided numerator and denominator by 1 - s.)
    q_n+1 = [1 / [(1/q _n - 1 + 1- s)/(1 - s)]                         (Subtracted 1 and added 1 to a term in the denominator.)
    q_n+1 = [1 / [[(1/q _n - 1) + (1- s)]/(1 - s)]                    (Regrouped terms.)
    q_n+1 = [1 / [(1/q _n - 1)/(1 - s) + (1- s)/(1 - s)]             (Distributed (1 - s) term.)
    q_n+1 = [1 / [(1/q _n - 1)/(1 - s) + 1]
    q_n+1 = [1 / [1 + 1 /{(1 - s) * (1/q _n - 1)}]
 
Applying the above equation to q₁ in terms of q₀:
    q₁ = 1/[1 + 1/(1 - s)*(1/q₀ - 1)]

Applying the same equation to q₂ in terms of q₁:
    q₂ = 1/[1 + 1/(1 - s)*(1/q₁ - 1)]

Plug the equation for q₁ into the equation for q₂:
    q₂ = 1/[1 + 1/(1 - s)*(1/{1/[1 + 1/(1 - s)*(1/q₀ - 1)]}- 1)]
    q₂ = 1/[1 + 1/(1 - s)*([1 + 1/(1 - s)*(1/q₀ - 1) - 1)]
    q₂ = 1/[1 + 1/(1 - s)*([1 + 1/(1 - s)*(1/q₀ - 1) - 1)]                (The -1 and 1 cancel.)
    q₂ = 1/[1 + 1/(1 - s)*( 1/(1 - s)*(1/q₀ - 1)]
    q₂ = 1/[1 + 1/(1 - s)*1/(1 - s)*(1/q₀ - 1)]
    q₂ = 1/[1 + 1/(1 - s)² * (1/q₀ - 1)]

We can see that if we continue this process,
    q_n = 1/[1 + 1/(1 - s)ⁿ * (1/q₀ - 1)]
 
 A slight change of notation yields Haldane's  equation for q_n:
   q_n = [1 + (1 - s)^-n * (q₀^-1 - 1)]^-1